Vector Spaces and Subspaces. Construction of Subspaces. The Initial Value Problem. Closed Form Solutions by the Direct Method.
Solutions Using Matrix Exponentials. Linear Normal Form Planar Systems. Formulas for Matrix Exponentials. Sinks, Saddles, and Sources. Phase Portraits of Nonhyperbolic Systems. Determinants and Eigenvalues. Existence of Determinants. Linear Maps and Changes of Coordinates. Linear Mappings and Bases. Least Squares Fitting of Data.
Autonomous Planar Nonlinear Systems. Saddle-Node Bifurcations Revisited. Proof of Jordan Normal Form. Higher Dimensional Systems. Linear Differential Equations. Solving Systems in Original Coordinates. Linear Differential Operators. Undetermined Coefficients. Periodic Forcing and Resonance.
The Method of Laplace Transforms. Laplace Transforms and Their Computation. Nonconstant Coefficient Linear Equations. Variation of Parameters for Systems. Simplification by Substitution. Exact Differential Equations. Numerical Solutions of ODEs. A Description of Numerical Methods. Local and Global Error Bounds.
Martin Golubitsky and Michael Dellnitz. The characteristic polynomial of the matrix is. An eigenvalue of is a root of the characteristic polynomial. There are three possibilities for the two eigenvalues of a matrix that we can describe in terms of the discriminant: The eigenvalues of are real and distinct.
The eigenvalues of are a complex conjugate pair. If it has an inverse, its rank is 8. So it has 8 eigenvectors I think? It doesn't matter whether matrix is invertible or not.
Although if a matrix is invertible then it means it is full rank i. This matrix has only one linearly independent eigen vector. Add a comment. Active Oldest Votes. For any of this, it doesn't matter whether or not the eigenvalues are non-zero. So all vectors other than the basis vectors would become linearly dependent. If there are eigenvalues, there are eigenvectors. Also, splitting field of the matrix. For example if the rotation is by 90 degrees then you are right there are no real eigen values but there are two complex eigen values.
In fact ,practically complex numbers as eigen value is not of much use but they do exist atleast mathematically. Show 2 more comments. Well, to be more precise: It depends on the underlying field Stefan Perko Stefan Perko Take e.
Hence regarded as a matrix over R its characteristic polynomial is a product of linear factors. Since to a real eigenvalue of a real matrix there always corresponds a real eigen-vector, repeating everything in the above for this special case of a real symmetric A with F as R , we obtain the following:. Spectral Theorem for Real Symmetric Matrices. If A is a real matrix, there exists a real orthogonal matrix U i. Note a real hermitian matrix is just a real symmetric matrix, and, a real orthogonal matrix a real unitary matrix.
Prove that A is normal iff its hermitian and skew-hermitian parts commute. Deduce that a hermitian, or a skew-hermitian matrix is normal. If a matrix equals its skew-hermitian part show that the matrix can have only purely imaginary eigenvalues. Does the converse hold? Inter-relate the hermitian and skew-hermitian parts of A and iA. If a real matrix A is unitarily diagonalizable and its eigenvalues are real then it is orthogonally diagonalizable, i.
If a matrix A is orthogonally diagonalizable then its eigenvalues are real iff A is real. A real matrix A is orthogonally diagonalizable iff its eigenvalues are real and it is normal. Let A be a real matrix with real eigenvalues. Let C be a non-singular complex matrix such that C -1 AC is diagonal. The size constraints make the equation meaningful. The set of eigenvalues of a matrix A is denoted by s A and is called the spectrum of A. Here b 11 is also the eigenvalue of B. Hence it is enough to assume the result for the sizes m-1 and n-1 and prove it for m, n.
Since the field of complex numbers is algebraically closed, all complex matrices are triangulable. Let Y triangulize A and Z, B. Then we can write. Hence either the system has no solution or more than one solution, completing the proof. Since only the triangulation of A and B has been used in the above proof, the result is valid for an arbitrary field over which the characteristic polynomials of A and B completely split i.
In particular the result is valid for all such systems over an algebraically closed field. Moreover, in the above, for infinite fields more than one solution means an infinity of solutions. An alternate proof of the above result may be based on an an easily verifiable fact that a polynomial in a triangulable matrix is singular iff the polynomial vanishes at least for one eigenvalue of the matrix. If A and B are not necessarily triangulable, a generalization of the above result is as follows a formal proof is left as an exercise for the reader :.
Note that the subsequent rows of this matrix are obtained by once circularly shifting the entries of the previous rows to the right. Show that the column vectors of the DFT matrix form a basis of C n that diagonalizes every circulant. Show that the set of all circulants is closed under the operations of scalar multiplication, addition and matrix multiplication and consequently forms an algebra over C.
If L is non-singular, i. A l -matrix is also called a matric polynomial in the indeterminate scalar l. Thus A l could be regarded as a polynomial in l with coefficients as matrices A 0 , A 1 , A l is called proper if the highest degree coefficient matrix A p is non-singular. The rank of a l -matrix A l is simply the rank of A l regarded as a matrix over the field Q F [ l ].
If det A l is a non-zero scalar, i. Consequently A p B q and B q A p are non-zero matrices. Corollary degree reduction. Remainder Theorem for l -Matrices.
Proof : The uniqueness follows from Corollary 2, since, e. The existence follows from the long-hand left and right division process. Theorem Similarity via l -Matrices. Since det R l is a non-zero scalar, R l -1 is a l -matrix. Construct appropriate l -matrices to check if the matrices. If A l , B l are proper, show that A l B l is proper. The following three types of row column operations are called l -elementary row column operations on l -matrices, or just elementary operations, or transformations on l -matrices:.
The operations i - ii are the same as the elementary row column operations discussed earlier. The difference lies in only the operation of type iii where here instead of just a scalar multiple we allow a more general l -polynomial multiple of a row column to be added to another.
A matrix obtained from the identity matrix by performing a l -elementary row column operation would be called an elementary l -matrix. It is interesting to note that every elementary l -matrix can be obtained by both a l -elementary row operation and a l -elementary column operation on an identity matrix. For, the three types of elementary l-matrices are:. Type i l -elementary matrix Type ii l -elementary matrix Type iii l -elementary matrix.
While adding a l times the j-th row to the i-th row of I results in the same matrix as obtained by adding a l times the i-th column to the j-th column.
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